Drawbacks of organ transplantation include the continuous requirement for immunosuppressive drugs and the occurrence of rejection episodes that, acutely or over time, contribute to the loss and/or decrease in the function of the allograft. In addition, the continuous requirement for immunosuppression is responsible for an increased incidence of infections and malignancies in long term transplant recipients. The investigators' central hypothesis is that it is possible to induce donor specific unresponsiveness in recipients of allografts by high-dose peripheral infusion of donor bone marrow cells (DBMC), without radiation conditioning of the recipients. In the absence of cytoablation, variables such as dose and timing of the DBMC infusions may constitute critical factors that affect graft survival. The investigators have strong preliminary evidence indicating that: 1) Human vertebral body bone marrow can provide a significantly higher dose of hematopoietic precursors compared to bone marrow from other sources, or peripheral blood. 2) Two or more DBMC infusions were safe as indicated by improved patient survival. 3) Two or more DBMC infusions resulted in improved liver allograft survival. 4) The investigators do not have any evidence that peripheral microchimerism is responsible for the improved results in the multiple DBMC infusion groups. Based on preliminary results, the investigators propose a prospective randomized clinical trial to determine the effect of two high-dose DMBC infusions on the incidence of rejection episodes and on the ability to withdraw immunosuppression in recipients of liver allografts. Even though it is not the investigators' central hypothesis that peripheral microchimerism is the determining factor in tolerance induction, they will continue to characterize by flow cytometry and flow-PCR the cellular composition of chimerism that follows DBMC infusions in recipients of liver allografts, and will retrospectively determine whether a correlation between the (quantitative and/or qualitative) characteristics of peripheral chimerism and post-transplant course (rejection and requirements for immunosuppression) exists. The development of strategies to improve long term allograft survival and/or the potential for progressive lowering and even discontinuation of immunosuppressive therapy would represent a substantial benefit to the patients and would also decrease the cost of post-transplant patient care.